3 The Utility Maximization Problem

3

The Utility Maximization Problem

We have now discussed how to describe preferences in terms of utility functions and how to formulate simple budget sets. The rational choice assumption, that consumers pick the best a¤ordable bundle, can then be described as an optimization problem. The problem is to …nd a bundle (x₁; x₂); which is in the budget set, meaning that

p1x1+ p2x2 m and x1 0; x2 0;

which is such that u(x1; x2) u(x1; x2) for all (x1; x2)in the budget set.

It is convenient to introduce some notation for this type of problems. Using rather standard conventions I will write this as

max x1;x2 u (x1; x2) subj. to p1x1+ p2x2 m x1 0 x2 0 ;

which is an example of a problem of constrained optimization.

A common tendency of students is to skip the step where the problem is written down. This is a bad idea. The reason is that we will often study variants of optimization problems that di¤er in what seems to be small “details”. Indeed, often times the di¢ cult step when thinking about a problem is to formulate the right optimization problem. For this reason I want you to:

1. Write out the “max”in front of the utility function (the maximand, or, objective func-tion). This clari…es that the consumer is supposed to solve an optimization problem. 2. Below the max, it is a good idea to indicate what the choice variables are for the

consumer (x1 and x2 in this example). This is to clarify the di¤erence between the

variables that are under control of the decision maker and variables that the decision maker has no control over, which are referred to as parameters. In the application

3. Finally, it is important that it is clear what the constraints to the problem are. I good habit is to write “subject to”or, more concisely, s.t. and then list whatever constraints there are, as in the problem above.

3.1

Solving the Utility Maximization Problem

Optional Reading: You may want to look at the appendix to chapter 5 in Varian (pages 90-94 in the most recent Edition).

We seek to solve the consumer problem

max x1;x2 u (x1; x2) subj. to p1x1+ p2x2 m x1 0 x2 0 :

Given that we are willing to assume that preferences are monotonic (which we are) we …rst make the simple observation that we may replace the inequality

p1x1 + p2x2 m

with

p1x1+ p2x2 = m:

To understand this one just needs to draw a graph. Suppose we would be able to …nd sine optimal solution x = (x₁; x₂)to the consumer problem such that p1x1+ p2x2 < mand that

preferences are monotonic. Then we observe that we can increase good 1 (or good 2 or both) a little bit without violating the budget constraint. But, by the monotonicity, this increases the utility of the consumer, which means that x wasn’t optimal. Hence we conclude that no optimal solution can be interior in the budget set.

6 -@ @ @ @ @ @ @ @ @@ s x1 x2 Better Bundles

Figure 1: Interior Bundles Can’t be Optimal with Monotonic Preferences

Thus, we can rule out all interior points in the budget set and solve max x1;x2 u (x1; x2) subj. to p1x1+ p2x2 = m x1 0 x2 0 :

But since the constraint must hold with equality we know that in any optimal solution it must be the case that

x2 =

m p1x1

p2

:

We can thus simply plug in the constraint into the objective function and solve the simpler problem max 0 x1_p1m u x1; m p1x1 p2 ;

WHICH IS A MAXIMIZATION PROBLEM WITH A SINGLE VARIABLE (exactly on the form maxxf (x) s.t. a x b which we had in our discussion on maxima). Ignoring for

now the possibility of corner solutions we can just di¤erentiate this and set the derivative to zero to get the …rst order condition.

3.1.1 Optimality Conditions for Consumer Choice Problem

Using the chain rule, di¤erentiate the utility function with respect to the choice variable x1

to get @u x₁;m p1x1 p2 @x1 + @u x₁;m p1x1 p2 @x2 p1 p2 = 0

Since the budget constraint is satis…ed with equality we can now substitute back x2 =

m p1x1

p2 ;

which leads to the condition

@u(x₁;x₂) @x1 @u₍x₁;x₂₎ @x2 = p1 p2 :

This condition, which we interpret as a tangency condition below, is a necessary condition for an interior optimum.

3.2

Interpretation of The Optimality Condition

The optimality condition has a useful geometric interpretation as a tangency condition be-tween the indi¤erence curve and the budget line and if often referred to as saying that

slope of indi¤erence curve=MRS=slope of budget line

This geometric interpretation is useful since it will allow us to go back and forth between pictures and math.

The …rst thing to realize is that

@u₍x0 1;x02) @x1 @u(x0 1;x02) @x2

is the slope of the indi¤erence curve for any point (x01; x02)(Remark about arguments &

notational sloppiness). To see this, pick a point (x0

1; x02) and look for all (x1; x2) such that

the consumer is indi¤erent between these points and (x0

1; x02) : That is

u (x1; x2) = u (x01; x02) k

| {z }

In principle, this can be solved for x2 as a function of x1 (as when we solved examples in

class & homework). This solution is some relation x2 = y (x1) satisfying

u (x1; y (x1)) = k

[I’m cheating here in that I’m ignoring some deep math....there is a question of whether u (x1; x2) = k can be solved for x2 as a function of x1]

Once you’ve taken on faith that we can solve for a function y (x1) we have that the

condition above is an identity ( holds for all values of x1). Hence, we can di¤erentiate the

identity with respect to x1 to get

@u (x1; y (x1)) @x1 +@u (x1; y (x1)) @x2 dy (x1) dx1 = d dx1 k = 0 m dy (x1) dx1 | {z } Slope = @u(x1;y(x1)) @x1 @u(x1;y(x1)) @x2 =MRS 0 @x1; y (x1) | {z } =x2 1 A

Now y (x1) is constructed so that for each x1 you plug in you get the same utility level (it

is an indi¤erence curve). So the interpretation of the slope of y (x1) is that it tells you how

much of good 2 you need to take away if you increase the consumption of good 1 slightly in order to keep the consumer indi¤erent.

6 -x2 x1 @ @ @ @ @ @ @_@ s y x + slope: @u(x1;y(x1)) @x1 @u(x1;y(x1)) @x2

Figure 2: The Marginal Rate of Substitution

the optimality condition can be interpreted as (after multiplying the optimality condition by 1) MRS (x1; x2) = @u₍x₁;x₂₎ @x1 @u(x₁;x₂) @x2 | {z }

rate at which the consumer is willing to exchange goods

= p1

p2

|{z}

rate at which the market is willing to exchange goods

3.2.1 Example: Cobb-Douglas Utility

u (x1; x2) = xa1xb2: Utility maximization problem:

max

x1;x2

xa₁xb₂ subj to p1x1+ p2x2 m

Budget constraint must bind)solve out x2 from constraint to get problem.

max

0 x1 _p1m

xa₁ m p1x1 p2

b

Assuming that the solution is not at the boundary points, the …rst order condition needs to be satis…ed. That is axa 1₁ m p1x1 p2 b + xa₁b m p1x1 p2 b 1 p1 p2 = 0:

At this point “only”some algebra remains. This can be done in a number of di¤erent ways, but here is one calculation

axa 1₁ m p1x1 p2 b + xa₁b m p1x1 p2 b 1 p1 p2 = 0 , multiply with 1 x₁ p2 m p1x1 b > 0 , m a x1 bp2 m p1x1 p1 p2 = 0 m a (m p1x1) bp1x1 = 0 , x1p1(a + b) = am , x1 = a a + b m p1

Now we have the candidate solution for good one. Plugging back into budget constraint gives m = p1x1 + p2x2 = p1 a a + b m p1 + p2x2 m m a a + bm = m a + b a a + b = b a + bm = p2x2 , x2 = b a + b m p2

Hence, the candidate solution is

x₁ = a a + b m p1 x₂ = b a + b m p2 Now,

We know that a solution must either satisfy the …rst order condition (the point (x₁; x₂) is the only point on budget line which does) or be at a boundary point.

In this example, any bundle with either x1 = 0 or x2 = 0 gives utility u (x1; x2) = 0;

whereas u (x1; x2) = (x1)a(x2)b > 0 since x1 > 0 and x2 > 0:

Combining these two facts we know that the bundle (x1; x2) = a+ba m p1; b a+b m p2 is indeed

the solution to the consumer choice problem.

3.2.2 Final Remark About Ordinality and Monotone Transformations

Let c = a a + b ) 1 c = 1 a a + b = a + b a a + b = b a + b

Plug in c instead of a and 1 cinstead of b above and you immediately get that

x₁ = cm p1 = a a + b m p1

so the solution does not change. This is because f (u) = ua+b1

is an increasing function of u and xa₁xb₂ 1 a+b _{= x} a a+b 1 x b a+b 2 = x c 1x 1 c 2

therefore is a monotone transformation. Hence there is no added ‡exibility in preferences by allowing a + b 6= 1 (and this simpli…es calculations). Thus, I will typically use utility function

u (x1; x2) = xa1x 1 a 2

Moreover, those of you who are comfortable with logaritms may note that ln xa₁x1 a₂ = a ln x1+ (1 a) ln x2

and you are free to use that transformations whenever you like. The FOC then immediately becomes a x1 + 1 a m p1x1 p2 p1 p2 = 0;

so this saves some work.

3.2.3 The Possibility of Corner Solutions

6 -a b x y f (x) f (b)

Recall that if the highest value of the function is achieved at a boundary point, then the …rst order condition need not necessarily hold at the maximum. I’ll skip the details, but it is straightforward to work out (and intuitive) that for our utility maximization problem:

1. If solution is at the lower end of the boundary (with x1 being the choice variable in

the reduced problem you get after plugging in the budget constraint), then that means that the slope of the indi¤erence curve at point (x1; x2) = 0;_pm

2 is ‡atter than the

budget line.

2. If solution is at the upper end of the boundary, then that means that the slope of the indi¤erence curve at point (x1; x2) = _pm₁; 0 is steeper than the budget line.

6 -x1 x2 c c c c c c c c c c c m p2 m p1

Figure 4: A Corner Solution

To see that this is a real possibility in a simple example, let u (x1; x2) = px1 + x2:

Following the same steps as in the previous example this results in a utility maximization problem that may be written as

max 0 x1 _p1m p_x 1+ m p1x1 p2 :

If there is an interior solution, we thus have that the …rst order condition (if f (x) = px then df (x)_dx = 1

must hold for some 0 x1 m_p

1: Solving this condition for x1 we get

x₁ = p2 2p1

2

:

For simplicity, set p2 = 2 and p1 = 1; in which case x1 = 1: Then, we can recover the

consumption of x2 through the budget constraint as

p1x1+ p2x2 = 1 + 2x2 = m)

x₂ = m 1

2 :

The point with this is that we notice that if m < 1; then our candidate solution for x₂ is negative, which doesn’t make any sense at all. We thus conclude that the solution must be at a corner and (it may be useful for you to plot the function px1 + 1 x₂1 to see this) the

solution is instead to use all income on x1;i.e.

(x₁; x₂) = (m; 0) :

solves the problem. If you have a hard time to see this you should 1) plotpx1+1 x₂ 1;and/or

2) plug in (x1; x2) = (m; 0) and note that u (m; 0) = pm; 3) plug in (x1; x2) = 0;m₂ and